Increasing the air-gap magnetic flux density to generate a higher torque in permanent magnet synchronous machines

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Master’s Thesis Lappeenranta University of Technology

School of Energy Systems Industrial Electronics

Bikarna Pokharel

INCREASING THE AIR-GAP MAGNETIC FLUX DENSITY TO GENERATE A HIGHER TORQUE IN PERMANENT MAGNET SYNCHRONOUS MACHINES

Examiners: Associate Professor Pia Lindh Professor Juha Pyrhönen

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ABSTRACT

Lappeenranta University of Technology School of Energy Systems

Industrial Electronics

Bikarna Pokharel

INCREASING THE AIR-GAP MAGNETIC FLUX DENSITY TO GENERATE A HIGHER TORQUE IN PERMANENT MAGNET SYNCHRONOUS MACHINES

Master’s thesis, 2017

86 pages, 41 figures, 11 tables, and 6 appendices

Examiners: Associate Professor Pia Lindh Professor Juha Pyrhönen

Keywords: Air-gap magnetic flux density, torque, permanent magnet synchronous machine (PMSM), permanent magnet (PM), Halbach array, PMSM rotor types

The main objective of this thesis is to investigate the air-gap magnetic flux density and the torque output of a permanent magnet synchronous motor with embedded magnet rotors.

In the beginning, variation of the air-gap magnetic flux density as a function of the magnet height has been investigated. Later, two rotor geometries have been investigated. One of the rotors has a dove-tail-shaped arrangement of magnets while the other has a U-shaped arrangement of magnets. The U-shaped magnets are oriented such that a Halbach array could be produced.

Two-dimensional Finite Element Analysis was employed in the rotor geometry analysis. The results of variation in the height of magnet suggest that air-gap magnetic flux density saturates after certain height of the magnet. Similarly, the results show that the embedded magnets in the rotor with Halbach array orientation could produce a higher air-gap magnetic flux density in comparison to the embedded magnets in the rotor with only radial orientation of magnets.

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ACKNOWLEDGEMENTS

The thesis was conducted as a part of project called AdMa project in the Electrical Engineering Department of Lappeenranta University of Technology.

I would like to first thank Professor Pia Lindh for supervising and guiding me to complete my thesis project.

I would then like to thank my second supervisor and examiner Professor Juha Pyrhönen for taking his time to examine my thesis pointing out the necessary changes.

I would also like to thank D.Sc. Janne Nerg for helping me with the configuration of the software, Cedrat Flux.

I am also grateful to my friends for comforting me during the duration of thesis and motivating me to complete my thesis.

Finally, I express my special gratitude to my parents for their support and understanding.

Lappeenranta, 2017 Bikarna Pokharel

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List of symbols and abbreviations

Roman Symbols

A linear current density

B magnetic flux density

Br remanent flux density

Dr diameter of rotor

F Force

H magnetic field strength

HcJ coercivity

is stator current

isd direct-axis stator current isq quadrature-axis stator current J

Lb

current density base inductance

Lm magnetizing inductance

Lmd, Ld direct-axis magnetizing inductance Lmq, Lq quadrature-axis magnetizing inductance

Lrσ rotor leakage inductance

Lsσ stator leakage inductance

Rs stator resistance

Sr surface area of rotor

T Torque

us stator supply voltage

EPM permanent-magnet induced back-emf iPM permanent magnet as current source

iD direct-axis damper current

iQ quadrature-axis damper current

RD direct-axis damper resistance RQ quadrature-axis damper resistance

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Greek Symbols

µrPM relative permeability of permanent magnet

δ load angle

θr rotor angle with respect to the stator coordinate system

μ0 permeability of vacuum

μr relative permeability

ρ resistivity

σFtan tangential stress

Φ power factor angle

Ψd direct-axis flux linkage

Ψmd direct-axis magnetizing flux linkage Ψmq quadrature-axis magnetizing flux linkage

Ψq quadrature-axis flux linkage

Ω electrical angular frequency

Subscripts

max maximum

n normal

tan tangential

Abbreviations

AC alternating current

AlNiCo Aluminium-nickel-cobalt

FEA Finite Element Analysis

FEM Finite Element Method

IM induction Motor

NeFeB Neodymium-iron-boron

PM Permanent Magnet

PMSM Permanent Magnet Synchronous Machine

pu per unit

RMS root-mean-square

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SmCo Samarium-cobalt

TC Tooth-coil

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1. INTRODUCTION

The objective of this thesis is to investigate the air-gap magnetic flux density and hence the torque output of a permanent magnet synchronous motor (PMSM). Several magnet geometries are studied in order to find out how to gain maximum air gap flux density for embedded magnet cases. Two separate rotor geometries were selected for detailed investigation, and the geometries were optimized to find out the change in the air-gap magnetic flux density of the machine. Furthermore, the maximum achievable torques with those geometries were also computed.

The two rotor geometries were constructed such that the magnet arrangements in one of them resembles a dove-tail and is built of five embedded magnet pieces, and the other resembles a U- shape with three magnet pieces. The magnets in case of the dove-tail arrangement were arranged to provide as high radial flux towards the stator of the machine. The magnets of the U-shape arrangement were oriented to obtain a Halbach array. More about Halbach array could be found from subchapter 2.4.2 of this thesis.

The major outcomes of this thesis are limited to the air-gap magnetic flux density and the torque analysis of the machine. The computations and investigations are performed for a 12-pole permanent magnet synchronous motor. Since this thesis has been carried out as part of a project done for a client, detailed information regarding the machine remains confidential.

1.1. Background

Electrical machines play an important role in the modern society. They find numerous applications worldwide. (Petrov & Pyrhönen, 2013a) Moreover, a report by the U.S. Department of Energy in 1980 reveals that medium power motors ranging from 0.75 kW to 90 kW consumed about 36% of the total electricity generated in the USA. This consumption accounts for about 60% of the electricity supplied to the motors in total (Dorrel, 2014) At present, 40% of the total electricity produced worldwide is consumed by electrical machines in industrial applications (De Almeida, et al., 2014).

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11 Inductions motors (IMs) still hold the designation “workhorse” of industry. Their robust rotor construction and mature manufacturing technology are the reasons behind gaining so much popularity in the industry. However, asynchronous machines suffer significant Joule losses in the windings of the stator and rotor. In order to reduce these losses, additional design actions should be implemented, many of which increase the overall manufacturing cost. For instance, the additional actions may include a larger sized machine, a copper-cage winding instead of aluminium rotor winding, high-performance steel that has a high magnetic permeability and low specific loss. (Petrov & Pyrhönen, 2013a) Despite the fact that these selections lead to a higher price of the electrical machine (Parasiliti, et al., 2004), they still are not able to solve the inherent drawbacks of asynchronous machines, which are the requirements for an additional proportion of current in the stator (magnetizing current component) and a slip-based-loss-generating current flowing on the rotor side.

When comparing the losses of an asynchronous machine with that of a permanent magnet synchronous machine (PMSM), it can be seen that the stator resistance losses in PMSMs are typically lower compared to the induction machines (IMs) as far as the power factor of the PMSM is better than that of the IM. In addition, PMSMs are free from rotor resistive losses since permanent magnets (PMs) are used in rotors instead of windings. Therefore, if other losses such as converter losses, iron losses, and friction losses are kept at the same level in both the induction machines and permanent magnet synchronous machines (PMSMs), it is possible to increase the efficiency of an electrical machine implementing permanent magnet (PM) technology. (Petrov & Pyrhönen, 2013a)

A significant drawback of PMSMs in industrial applications is the fluctuating price of the rare- earth materials, i.e. neodymium and dysprosium, which, in addition to iron, are the fundamental components of modern high performance permanent magnets (Morimoto, et al., 2014). The rapid increase in the cost of rare-earth magnets in 2011 boosted research on rare-earth-free electrical machine solutions in applications such as automotive electric propulsion systems (Boldea, et al., 2014). Some of the proposed approaches for rare-earth-free electrical machines which can be found in literature are – a synchronous reluctance ferrite permanent magnet assisted machine (Obata, et al., 2014) (Takeno, et al., 2011), a switched reluctance machine

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12 (Chino, et al., 2011), a ferrite spoke-type rotor PMSM (Kim, et al., 2014), an outer rotor ferrite PMSM (Petrov, et al., 2013b), and some other interesting ideas.

An alternative approach to rare-earth permanent magnets in electrical machines is the use of ferrite permanent magnets. The ferrite permanent magnets are abundant and their cost is comparatively low. However, they suffer from low remanent flux density and a higher demagnetization risk when compared to the rare-earth permanent magnets (PMs).

1.2. Design of a Permanent Magnet Synchronous Machine (PMSM)

In a rotating electrical machine, torque production is of interest. Electromagnetic torque analysis is essential for machine design and control (Wang, et al., 2014). According to Maxwell’s stress- tensor equation, the tangential stress in the machine is increased as the air-gap magnetic flux density is increased, which in turn increases the torque of the machine (Pyrhönen, et al., 2014a).

An intuitive approach for higher air-gap flux density is the elimination of the air-gap. However, a physical air-gap is essential for the machine to rotate. Andy Judge (2012) has proposed the use of Ferro-fluids to eliminate the air-gap during his studies on “Air-gap elimination in permanent magnet machines”.

Rotor magnetic arrangement is a vital step in the design of permanent magnet synchronous machines. Different kinds of rotor magnet arrangements such as buried tangential, radial, and inclined magnets exist in addition to their commonly used rotor-surface-mounted counterparts.

The Halbach array configuration has traditionally been used to produce harmonically pure magnetic fields in undulators and electron wigglers. An ideal Halbach array configuration is the one whose magnetization varies continuously in a sinusoidal fashion. (Ofori-Tenkorrang &

Lang, 1995)

Permanent magnet (PM) materials are widely used in electrical machines. The permanent magnet machines find their applications, for instance, in industrial machines, wind power generators, traction motors, linear machines, high-speed machinery, and machines used in aerospace applications. (Lindh, et al., 2014) Comparison of various types of electrical machines has resulted in that the permanent magnet motors are the best candidates for applications like

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13 aerospace, high-precision process control, marine navigations, military etc. that require high torque and high power density with very fast dynamic response capability (Wang, et al., 2006) Sintered magnetic materials have a low macroscopic resistivity, which can vary in the range of 100-200 µΩcm, providing eddy currents with paths to run and produce losses. In some cases, when bulky sintered magnets are assembled on the rotor surface, the eddy current losses can be so high that the polarization of the magnetic material is lost because of a high operating temperature in the magnets. Magnetic segmentation is a suggested method to limit the eddy current losses in the same way as in magnetic steel cores of the machines made of laminations.

(Pyrhönen, et al., 2014b) Figure 1 illustrates the flux losses experienced by a neodymium-iron- boron (NdFeB) magnet at elevated temperatures.

Figure 1 An illustration of flux losses in a NdFeB magnet at elevated temperatures. It shows that the magnetic moment of the PM materials at higher temperatures goes to a region of unrecoverable irreversible losses. Redrawn based on the information from (Masuzawa, et al., 2006)

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14 In carefully designed rotating field permanent magnet machines, the permanent magnet (PM) material operates in the second quadrant of the magnetization curve, and it could be concluded that the hysteresis losses cannot play an important role in them. However, in permanent magnet machines having a strong armature reaction, part of the magnets at certain loads, may operate at flux densities where hysteresis losses can be present. (Pyrhönen, et al., 2014b)

Marinescu and Marinescu (1992) compared the air-gap flux of a discrete Halbach array where the magnetization vector rotates 60 degrees, i.e. 3 magnet segments per pole, and 45 degrees, i.e. 4 magnet segments per pole between magnet pieces to that of a conventional (radially magnetized) rotor-surface-mounted arrangement; the result of which showed that the multipolar Halbach array is capable of producing a higher peak air-gap flux and hence higher machine torque than an ordinary arrangement.

Ofori-Tenkorrang & Lang (1995) have concluded that permanent magnet motors where the application precludes the use of a magnetic yoke iron, the Halbach array always produces higher torques than a conventional array for the same volume of magnets. However, for iron-yoke rotors, a conventional array with an optimized pole-arc to pole-pitch ratio produces a higher torque than a Halbach array up to a certain thickness of magnets. Above this thickness, the Halbach array produces a higher torque.

The shape and polarization direction of a rotor surface magnet pole has a significant impact on the air-gap magnetic flux density distribution, which cannot only vary the amplitude of the fundamental air-gap magnetic flux density, but also produce torque ripples and therefore induce vibration. Hence, for an optimized electromagnetic performance of the machine, numerous rotor pole designs have been proposed over the past decades, such as Halbach magnetization (Zhu &

Howe, 2001), magnet segmentation (Ashabani & Mohamed, 2011), pole shaping (Zhu, et al., 2012), and modular pole (Isfahani, et al., 2009) among others.

In a study by (Shen & Zhu, 2012), they have proposed magnetic arrangements that combine both Halbach magnetization as well as unequal magnet heights as depicted in Figure 2 that shows the H-type and T-type magnetic pole configurations. This ensures a low torque ripple and uses a low amount of magnet material, but avoids a reduction of the fundamental air-gap magnetic flux density and corresponding torque due to inter-pole leakage flux. They concluded

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15 that a T-type magnet pole could exhibit significant potential to achieve a low torque ripple, a high electromagnetic torque, and a low magnet material amount, and it presents a larger ratio of average output torque to permanent magnet volume in contrast with conventional three-segment Halbach array. T-type pole implementation also reduced inter-pole leakage flux significantly in comparison to the H-type magnet pole and traditional magnet-shaping techniques.

Figure 2 Schematic configuration of H-type (a) and T-type (b) magnetic poles. Redrawn based on the information from (Shen & Zhu, 2012)

In a study by (Gundogdu & Komurgoz, 2010) for the design of 12-poled permanent magnet motors with different rotor types with NdFeB magnets and steel (Steel 1008) having saturation points at 1.23 T and 2.00-2.30 T respectively for six different motor types, the magnetic analysis showed results of no-load magnetic flux density, B, as shown in Table 1.

The different motor types are labelled a-f. The motor types a, b, and c respectively represent rotor geometries with rotor surface mounted magnets-vertical to shaft Figure 6(b), rotor surface mounted magnets-parallel to shaft Figure 3(i), and rotor surface mounted magnets Figure 3(ii).

Similarly, the motor types d and e represent rotor geometries with buried radial magnets Figure 6(f) and buried tangential magnets Figure 6(c) respectively. The motor type f is a permanent magnet machine with an outer rotor Figure 3(iii).

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16 Figure 3 Motor types: rotor surface mounted magnets-parallel to shaft (i), rotor surface mounted magnets (ii), and permanent magnet machine with an outer rotor (iii). Redrawn based on the information from (Gundogdu &

Komurgoz, 2010)

Table 1 Magnetic data (no-load magnetic flux density) of six different permanent magnet motor designs

Motor type a b c d e f

Magnetic data

Stator Teeth B [T] 2.03 1.89 1.93 2.05 0.49 2.09

Stator yoke B [T] 1.90 1.89 1.94 2.05 0.45 0.47

Rotor yoke B [T] 0.39 0.41 0.40 0.38 0.08 2.17

Air gap B [T] 1.16 1.07 1.10 1.16 1.03 0.96

Magnet B [T] 1.04 1.11 1.09 0.81 1.21 0.95

1.3. Methodology and Materials

Different published research papers, journals, and few literatures have been the basis for this thesis. Institute of Electrical and Electronics Engineers (IEEE) publications were reviewed at higher frequencies. The literature titled “Design of Rotating Field Machine”, second edition, by (Pyrhönen, et al., 2014a) was also very helpful in the development of this thesis. Finite element analysis (FEA) have been used for the investigations of air-gap magnetic flux density and the torque of the machine. A Finite Element Method (FEM) software, Cedrat Flux 2D, was used for

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17 all the investigations. In addition, Matlab, a mathematical software tool from Mathworks, has also been used to plot the curves obtained from different investigations.

1.4. Organization and Contents of Thesis

This thesis has been divided in six different chapters with number 1 to 6. The subsections of the chapters have been numbered on the level basis, for instance, subsections of chapter 1 are 1.1…, and 1.1.1…, etc. corresponding to sublevels 2 and 3 respectively. The contents of each of the chapters are briefly discussed as follows:

Chapter 1 presents an introduction followed by a background that contains information from different research papers, journal articles, and literatures. The role of rotor geometry and magnets in the design of Permanent Magnet Synchronous Machines (PMSMs) have also been presented in this section.

Chapter 2 presents the theoretical aspects of a permanent magnet synchronous machine (PMSM). This section and its subsections provide insights on cooling of PMSM, different rotor geometries of a PMSM, equivalent circuit and vector diagram of a PMSM, and a brief comparison between rotor-surface-magnet and embedded magnet PMSMS. Furthermore, a subsection is dedicated to permanent magnets (PMs), their history of evolution, their demagnetization characteristics, losses in them, and a special arrangement of magnets called the Halbach arrays.

Chapter 3 deals with the Maxwell’s stress-tensor and torque production theory. This chapter studies the main objective of this thesis, which was to find the maximum possible air-gap magnetic flux density in a PMSM so that the torque of the machine could be increased for a given amount of supplied current. The effect of temperature on the air-gap magnetic flux density and thus the torque production has been included in its subsections. In addition, a glimpse on potting materials used in rotating field machines for cooling purposes has also been presented.

Chapter 4 presents investigations on rotor geometries. The major investigations include the computations of air-gap magnetic flux density and the torque of the machine. General dependence of the air-gap magnetic flux density on the magnet heights have been presented in

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18 the beginning of this section. Then investigations presented in this section were based on two different embedded rotor geometries.

Chapter 5 gathers the major results obtained from different investigations and computations performed. This chapter essentially acts as a summary of chapter 4.

Chapter 6 has been dedicated for the conclusions of the investigations and computations performed during the project for this thesis. Moreover, all the major information, which are the outcomes of this thesis could be found in this chapter.

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2. PERMANENT MAGNET SYNCHRONOUS MACHINES

Permanent Magnet Synchronous Machines (PMSMs) fall under the family of synchronous machines. The PMSMs can be further classified on the basis of saliency as illustrated by Figure 4. They resemble closely any other kind of synchronous machines. They essentially differ from other kinds of synchronous machines in the rotor construction. The rotor of a PMSM has permanent magnets instead of field windings. For example, the vector diagrams of a Permanent Magnet Synchronous Machine (PMSM) and any other synchronous motors is the same except that in PMSMs, the magnets create their own constant magnetic flux linkage (𝛹_PM) instead of controllable magnetic flux linkage (𝛹_f), which is the case in other synchronous machines.

(Pyrhönen, et al., 2014a)

Permanent magnet excitation makes the design of the permanent magnet machine highly efficient since there are, in principle, no losses in the excitation. However, many permanent magnet materials are conductive in nature, thus making them a potential source of Joule losses.

Moreover, rare earth magnets are expensive, which can have a major effect in the design and construction of the machine. Electrical machines with high torque density are typically required by sectors for which the machine dimension and weight should be minimized. These include for instance wind power, truck, and hybrid drive sectors. (Polikarpova, 2014)

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20 Figure 4 Further classification of Permanent magnet synchronous machines (PMSMs) (Pyrhönen, et al., 2014a)

The Permanent Magnet Synchronous Machines (PMSMs) are further classified as salient PMSM and non-salient PMSM, which has been shown in Figure 4 above. The salient PMSMs are often characterized by their q-axis inductance (𝐿_q) being greater than their d-axis inductance (𝐿_d), i.e. 𝐿_q/𝐿_d > 1 whereas the non-salient PMSMs are characterized by their q-axis inductance and d-axis inductance being almost equal, i.e. 𝐿_q/𝐿_d ≈ 1. (Pyrhönen, et al., 2014a) Figure 4 above also shows that the salient PMSM can be further categorized as Permanent magnet embedded PMSM and pole-shoe PMSM. The pole shoe PMSM is essentially a rotor-surface- mounted magnet PMSM, i.e., non-salient PMSM that consists of pole shoes on the magnet poles to produce saliency.

The Permanent Magnet Synchronous Machines (PMSMs) can alternatively be divided into two classes depending upon the direction of field flux as follows:

 Radial flux PMSM: in this class of PMSMs, the flux lines in the air gap travel along the radius of the machine, hence the name radial field PMSM.

 Axial flux PMSM: The flux lines in the air gap travel parallel to the machine shaft in this type of PMSM.

Permanent Magnet Synchronous Machines (PMSMs)

Salient PMSM

Permanent magnet embedded PMSM

Pole-shoe PMSM

Non-salient PMSM

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21 Over the past decade, tooth-coil permanent magnet synchronous machines (TC PMSMs) have gained increasing popularity. The TC PMSM features a special winding system in which the stator windings are arranged in each or every second stator tooth. The tooth coil simplifies the manufacturing process in comparison to the traditional distributed windings. Moreover, this kind of a winding system contains very short end windings. This also results in shorter axial length of the machine, which in turn reduces the stator copper losses due to decreased copper volume. (Magnussen & Sadarangani, 2003) Figure 5 shows a three dimensional model of a typical Tooth-Coil Permanent Magnet Synchronous Motor (TC PMSM) with stator and rotor separately.

Figure 5 A three- dimensional model of a typical tooth-coil permanent magnet synchronous motor (Petrov &

Pyrhönen, 2013a) Reproduced with permission from Ilya Petrov.

2.1. Rotor of Permanent Magnet Synchronous Machines

The Permanent Magnet Synchronous Machines (PMSMs) are characterized by the construction of their rotor. The magnet placement in the rotor determines the saliency of the motor. For instance, a rotor with surface-mounted magnets is in principle a non-salient type as the relative permeability of magnet, for instance, Neodymium Iron Boron (NdFeB) magnet, is close to the permeability of vacuum, i.e. close to 1 (specifically 1.04-1.05), and permanent magnet embedded rotor produces without any exception with a quadrature axis inductance higher that the direct axis inductance. (Pyrhönen, et al., 2014a)

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22 Figure 6 below illustrates some of the rotors of PMSMs with different types of magnet placements and arrangements. Each of the geometries has its own significance. For instance, as shown in Figure 6(d), two magnets per pole in V-shape could produce a high air-gap magnetic flux density at no-load condition. Figure 6(g) above also shows a rotor geometry in which the magnets are arranged in a dove-tail fashion, which has been investigated in this thesis project.

Figure 6 Different types of magnet arrangements in the rotor of permanent magnet synchronous machines (PMSMs) (a) magnets embedded on the surface, (b) rotor surface magnets, (c) tangentially embedded magnets, (d) two magnets per pole arranged in V-shape, (e) pole-shoe rotor, (f) radially embedded magnets, and (g) magnets arranged in a dove-tail shape. (a) - (f) are redrawn based on the information from (Pyrhönen, et al., 2014a), and (g) is one of the rotor types investigated for this thesis.

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2.2. Rotor-Surface-Magnet and Rotor-Embedded-Magnet Permanent Magnet Synchronous Motors

Rotor-surface-magnet permanent magnet motors are the most common construction for PMSMs these days. In a rotor-surface-magnet PMSM, the magnets are often magnetized radially. In such motors, the d- and the q- axes’ inductances are considered to be equal, which in fact makes the design of rotor-surface-magnet PMSMs simpler. Furthermore, since the magnets are attached on the rotor surface, the design is easier and cheaper. (Lindh, et al., 2003)

Embedded magnet PMSMs on the other hand can have circumferentially magnetized permanent magnets embedded into deep slots of the rotor. This embedding of magnets produces saliency in the construction, which in turn makes the q-axis synchronous inductance greater than the d- axis synchronous inductance. In case the shaft of the motor is ferromagnetic, a significant proportion of the magnetic flux produced by the PMs can go through the shaft. (Lindh, et al., 2003)

An important advantage of the rotor-surface-magnet PMSMs over embedded-magnet PMSMs is that the amount of magnet materials needed to design, for example an integral slot machine, is smaller. For a machine with a same size when constructed as a rotor-surface-magnet or embedded-magnet machine, the same power can be achieved with the rotor-surface-magnet PMSM with a smaller amount of magnets than in the case of embedded-magnet PMSMs.

However, the use of embedded magnets in PMSMs are favoured by several other advantages.

The embedded magnet version of PMSMs may generate more torque per rotor volume compared to the rotor-surface-magnet mounted version owing to its high air-gap magnetic flux density.

This however, necessitates usually significantly more PM-material. PM demagnetization danger remains smaller in embedded PMSMs. The rotor-surface-magnet version of PMSMs may have fixing and bonding problems which is not the case in embedded-magnet versions where the magnets could be easily mounted into the pockets of the rotor. (Lindh, et al., 2003)

The load angle equation of a synchronous machine is also an important subject of analysis in case of a PMSM. The torque equation corresponding to the load angle equation is given with

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24 phase quantitites for voltage Us, permanent magnet induced voltage EPM, supply angular velocitys and d-and q-axis synchronous inductances Ld and Lq as:

𝑇 = 3𝑝 (𝑈_s𝐸_PM

𝜔_s²𝐿_d sin𝛿 + 𝑈_s² 𝐿_d− 𝐿_q

2𝜔_s²𝐿_d𝐿_qsin 2𝛿) (1) If the machine is non-salient type, the pull-out torque depends on the synchronous inductance inversely. This necessitates for a lower inductance for a high torque production ability of the machine. The per-unit version of (1) is got by removing 3p multiplicator and replacing all values with their pu values except the load angle . Often due to practical reasons, the per unit (pu) value of EPM has to be close to 1. As the supply voltage is also 1, the pu value of the synchronous inductance has to be selected at a value that is considerably below the value 1. If the machine is a non-salient pole type, and EPM = Us = 1 per unit, the synchronous inductance has to be Ld = 0.625 per unit at maximum, in order to achieve the commonly required 160 % pull-out torque.

When using rotor surface mounted magnets, this precondition of synchronous inductance to be relatively low is met quite easily. In machines with embedded magnets, the pu values approach 1, and therefore the per unit value of EPM cannot be increased considerably above one.

(Pyrhönen, et al., 2016)

In this thesis, investigations on embedded magnet permanent magnet synchronous motors have been presented with two different magnet arrangements in the rotor. One of the arrangements is embedded magnets arranged in a dove-tail fashion while the other is a special case of embedded magnets with magnets oriented to produce Halbach array. The investigations have been presented in Chapter 4 of this thesis.

2.3. Equivalent Circuit and Vector Diagram of PMSM

The equivalent circuit and vector diagrams are often used for control purposes in electrical drive systems. However, the objective of this thesis is limited to the investigation of the air-gap magnetic flux density and torque production in a PMSM. Figure 7 shows the equivalent circuit of a PMSM. The circuit is quite similar to separately excited synchronous machines except that the magnet is replaced by a current source (Pyrhönen, et al., 2014a). In synchronous machines,

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25 the equivalent circuit is divided into d- and q- axes equivalent circuits such that the investigations on the d- and q- axes characteristics, for instance d- and q- axes inductances, can be done separately. This again helps in the control aspect of the machine.

Figure 7 Equivalent circuits of a PMSM in the d-axis (upper) and q-axis (lower) directions. Redrawn based on the information from (Pyrhönen, et al., 2014a)

In Figure 7 above, the d-axis and q-axis supply voltages are denoted by 𝑢_d and 𝑢_q respectively.

Similarly, the d-axis and q-axis stator currents are denoted by 𝑖_sd and 𝑖_sq respectively. 𝜓d and 𝜓q respectively refer to the d-axis and q-axis synchronous flux linkages. 𝜓md and 𝜓mq represent the d-axis and q-axis magnetizing flux linkages respectively. 𝑅s, 𝑅D, and 𝑅Q respectively are the

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26 stator resistance, d-axis damper resistance, and q-axis damper resistance. In the same way, 𝐿sσ, 𝐿Dσ, and 𝐿Qσ respectively represent the stator leakage, d-axis damper leakage, and q-axis damper leakage inductances.

The permeability of magnets are very low (ideally µrPM = 1), and thus the magnetizing inductance (𝐿m) of permanent magnet machines usually becomes low. Moreover, the synchronous inductance per unit value must normally be lower than one per unit (1 pu) in order to produce enough peak torque at nominal voltage and speed. The reason for this is that the maximum torque is inversely proportional to the synchronous inductance. Equation (2) shows the load angle equation which is the same as the load angle equation for other synchronous machines except that 𝐸_f in case of permanent magnet machines is replaced by 𝐸PM. Equation (1) is the torque equation corresponding to equation (2).

𝑃 = 3 (𝑈_s𝐸_PM

𝜔_s𝐿_d sin𝛿 + 𝑈_s² 𝐿_d− 𝐿_q

2𝜔_s𝐿_d𝐿_qsin 2𝛿) (2) In the equation the parameters indicated by 𝑃, 𝑈s, 𝐸PM, 𝜔s, 𝐿d, 𝐿q, and 𝛿 respectively represent power, supply voltage, back-emf of the permanent magnets, stator angular frequency, d-axis synchronous inductance, q-axis synchronous inductance, and the load angle.

The vector diagram is also a modification of the vector diagram of a separately excited synchronous machine. Figure 8 represents the vector diagram for a PMSM drawn at its nominal operating point such that it operates as a motor with 𝑢s = 1, 𝑖s = 1, 𝜑s = 12^, and 𝑖sd = 𝑖sq = 0.5.

Generally, the vector diagram is represented in the rotor (dq) and stator (XY) reference frames.

The permanent magnets create the flux linkage (𝜓_PM) in the stator windings. (Pyrhönen, et al., 2014a)

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27 Figure 8 Vector diagram of a PMSM presented in the stator (XY) and rotor (dq) reference frames. 𝝍PM is the permanent magnet created flux linkage vector, 𝒖s is the supply voltage vector, which is integrated to produce stator flux linkage vector, 𝝍PM, 𝒊s is the stator current vector which is resolved into d- and q- axes components as 𝒊sd and 𝒊sq respectively, 𝒍sd and 𝒍sq are the d- and q- axes inductance vectors respectively, 𝝋s is the power factor angle, 𝜽r

is the angle of rotor with reference to the stator (XY) reference frame, and 𝜹s is the load angle. Redrawn based on the information from (Pyrhönen, et al., 2014a)

2.4. Cooling in Permanent Magnet Machines

Electrical machines with permanent magnets (PM) possess special requirements for the cooling system. In PMSMs, heat propagation from the stator windings to the permanent magnets should be minimized due to the temperature sensitiveness of the rare earth PM materials. So, the stator windings generated heat should be somehow removed via stator yoke, frame, or internally to keep the temperature of the PM lower than its demagnetization temperature. (Polikarpova, 2014) Conventional air cooling or forced air cooling is not adequate for permanent magnet machines with high density as air removes heat from the end windings and air gap, which also transfers heat to the magnets of the rotor. (Polikarpova, 2014) The performance of the cooling system

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28 must be improved with increasing torque density of the PM electrical machine. Bruetsch et al.

(2008) deduced that up to 73% of the damage to electrical machines is caused by over temperature. It is generally accepted that air cooling solutions are easier and more reliable than water cooling solutions. However, in some cases, cooling solutions for the highest power electrical machines should adopt direct or indirect liquid cooling to become more compact and thus more attractive (Polikarpova, 2014).

In general operating temperature range, the residual flux density and the intrinsic coercivity of magnets decrease with increase in temperature. However, decreasing the temperature will bring back the residual flux density and the intrinsic coercivity to the magnets normal values, and thus the change is reversible. In case, the temperature is increased further above the normal operating range, the magnets get demagnetized and this demagnetization is irreversible. The variation in the residual flux density along with the variation in armature resistance of the motor due to temperature influences the torque capability as well as the efficiency of the motor. (Sebastian, 2002)

Permanent magnet machines require special cooling system. In permanent magnet synchronous machines (PMSMs), the heat propagates from the stator windings to the permanent magnets, which must be minimized owing to the temperature sensitiveness of the rare-earth magnets.

(Polikarpova, 2014) A careful design is one, which does not bring about a permanent demagnetization in the magnets.

In recent years, there has been a growing interest in high thermal conductance material application such as high conductance potting materials (for instance, aluminium nitride, high performance epoxy, graphite foam, and thermoplastic) (Hoerber, et al., 2011). The application of high conductance material allows for the balancing of heat flux or redistribution of heat towards the cooling system, but it alone does not remove the heat (Polikarpova, 2014). Yao et al. (2011) applied a compound based on aluminium nitride with a thermal conductivity of 40 W/ (K·m) to the end winding of a 7 kW PMSM and achieved a reduction of 20 K in the maximum temperature.

Typical average tangential stress and linear current density values as a function of cooling solution is presented in Table 2. The values indicated in brackets are the values from the machine

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29 studied by (Polikarpova, 2014). Additionally, the fourth column of the table also illustrates the current density. At increased permanent magnet temperatures, the magnetic polarization decreases which leads to drop in the torque. This demands for an increase in the current density to meet the required torque.

Table 2 Typical average tangential stress and linear current density values as a function of cooling solution

Cooling Method Tangential Stress, [kPa]

Linear current density, [kA/m]

Current Density, J, [A/mm²] Air cooling (salient

pole)

 Passive

 Forced

<50…60 (< 30)

- (8.5)

<80 (<60)

- (38)

1.5-5

- 5-10 (3.4)

Hydrogen Cooling - 90-110 -

Liquid Cooling (Single-phase)

 indirect (water jacket)

 direct (immersion oil cooling, direct cooling through hollow strands)

>50

<60 (22, 33)

>60 (80)

70-200

90-100 (30, 48)

110-200 (130, 147)

7-30

7-10

10-30 (8, 4.8)

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30

2.5. Permanent Magnets

Permanent magnets (PMs) have been extensively used in synchronous machines to replace the excitation windings. The most widely used permanent magnets (PMs) in electrical machines are Alnico, Ceramics (Ferrite), and rare earth magnets, i.e. samarium cobalt (SmCo) and neodymium-iron-born (NdFeB) magnets. (Gundogdu & Komurgoz, 2010)

The development and manufacture trend of permanent magnets (PMs) is shown in Figure 9. The curve shows that the development and manufacture of PMs began only in the early twentieth century with the production of carbon, cobalt, and wolfram steels.

Figure 9 Development trend of Permanent magnet materials showing the materials - Steel, Ferrite, AlNiCo, SmCo5, 6. Sm2Co17, and NdFeB.The theoretical limit of magnetic energy (BH)max NdFeB is also shown in figure. In

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31 addition, furture possibilities of permanent magnet materials has been represented by the dashed-dotted curve.

Redrawn based on the information from (Dawwson, et al., 2014)

Permanent magnets are characterized chiefly by the following eight properties:

i. Remanence 𝐵_r,

ii. Coercivity 𝐻_cJ (or 𝐻_cB),

iii. The second quadrant of the hysteresis loop, iv. Energy product,

v. Temperature coefficients of remanence 𝐵_r and coercivity 𝐻_cJ, vi. Resistivity ρ,

vii. Mechanical characteristics, and

viii. Chemical characteristics (Pyrhönen, et al., 2014a)

Table 3 shows the comparison of different characteristics of present day neodymium-iron-boron (NdFeB), samarium-cobalt (SmCo), and ferrite magnets.

Table 3 Comparison between different magnetic characteristics of NdFeB, SmCo, and ferrite magnetic materials (Goldman, 2006). The values in brackets are in the column NdFeB are from (Arnold Magnetic Technologies, 2016), and the values in brackets in the column SmCo are from (Arnold Magnetic Technologies, 2017)

Characteristics NdFeB SmCo Ferrites

Remanence, 𝐵_r [T] 1.44 (1.49) 1.12 (1.17) 0.41

Coercivity, 𝐻_c [kA/m] 1115 730 (867) 240

Energy density, (𝐵𝐻)_max [kJ/m³] 400 (430) 240 (255) 32 Max cont. Temperature,

𝑇_max [℃ ]

80 – 160 (80 –

220) 300 (350) 250

Resisitivity, [(Ω-m)x10^-6] 1.1 – 1.7 0.65 – 0.9 10⁹ Relative permeability, µr 1.04 – 1.1 1.04 – 1.12 1.1 – 1.3

Neodymium magnets are the strongest and the most commonly used present-day rare-earth magnets. On the basis of manufacturing process, the magnets may be classified as sintered or

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32 bonded magnets. These are available in different shapes, sizes, and grades. The sintered magnets are stronger than the bonded magnets. (China Rare Earth Magnet Limited, 2004-2013)

The use of permanent magnets in electrical machines have the following advantages:

 No electrical energy is absorbed by the field excitation system and hence, in principle, there are no excitation losses, which means substantial increase in efficiency,

 higher torque and/or output power per volume than using electromagnetic excitation,

 higher magnetic flux density in the air-gap, thus better dynamic performance than the motors with electromagnetic excitation,

 simplified construction and maintenance, and

 reduction in the cost for some machine types. (Gieras & Wing, 2002)

2.5.1. Permanent Magnet Demagnetization

The demagnetization of permanent magnets (PMs) is the function of temperature. The second quadrant of the hysteresis loop of PM material is of significance from the design point of view (Pyrhönen, et al., 2014a). With an increase in temperature, the coercivity of magnet decreases and thus becomes susceptible to demagnetization. More about permanent magnet demagnetization curves can be found from the manufacturers’ websites.

2.5.2. Halbach Array

The Halbach array has traditionally been used to produce harmonically pure magnetic fields for use in undulators and electron wigglers (Ofori-Tenkorrang & Lang, 1995). In a study by (Winter, et al., 2012) for axial Halbach arrays using 3D Finite Element Analyses (FEA), they concluded that the Halbach magnetic arrangement is a beneficial choice for reducing the weight, increasing the flux density, and thus increasing the torque output of ironless axial flux motors.

As an example, a schematic presentation of Halbach array is shown in Figure 10. The arrows point in the direction of polarization of respective magnets. There are five magnets arranged to form a Halbach array such that they have different magnetic orientation so as to produce a concentrating magnetic field only on one side.

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33 Figure 10 A schematic diagram showing a simple Halbach array such that each piece of magnets has its own direction of orientation.

It has been shown by (Trumper, et al., 1993) that in an ideal Halbach array, the magnetic field intensity vanishes on one side of the array, and for a square block array such as shown in Figure 11, the spatial fundamental field is cancelled on one side of the array whereas the field intensity on the other side of the array is multiplied by a factor of √2 .

Figure 11 Halbach magnetic field intensity showing the intensity vanishing in the upper side and concentrating on the lower side. (Ofori-Tenkorrang & Lang, 1995)

The significances of using a Halbach array configuration over a conventional permanent magnet arrangement are listed as follows:

 Stronger fundamental field compared to conventional permanent magnet arrangements,

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34

 heavy-weight backing steel elements can be omitted and thus no iron losses occur,

 the magnetic flux density is more sinusoidal, and

 very low back side fields. (Gieras, et al., 2004)

2.6. Losses in Permanent Magnet Synchronous Machines

Permanent magnets are conductive, hence Joule losses take place in them (Pyrhönen, et al., 2014a) In PMSMs, even 80% of losses are generated in the stator. The rotor losses in a PMSM mainly include the rotor iron losses and the losses in the permanent magnets (PMs).

(Polikarpova, 2014). Eddy current losses in the rotor of a permanent magnet synchronous machine may take place due to three different reasons as listed below:

1. Stator winding produces a significant amount of current linkage harmonics that generate flux densities travelling across the permanent magnets, thereby causing eddy currents. These are also called winding harmonics.

2. The slot openings cause flux density variations that induce eddy currents in the permanent magnets. These are also called permeance harmonics.

3. Frequency – converter caused time harmonics in the stator current waveform causes extra losses in the rotor. (Jussila, 2010)

Hysteresis losses also can take place in addition to the eddy current losses when AC or a rotating flux travels through the permanent magnet materials (Pyrhönen, et al., 2014a).

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35

3. MAXWELL’S STRESS TENSOR AND TORQUE PRODUCTION

Maxwell’s stress tensor is often employed in numerical methods for the calculation of forces or torque. It is based on Faraday’s statement according to which stress occurs in the flux lines. The linear current density A on a metal surface creates tangential field strength components on the metal surface, and such tangential field strength components are essential for both tangential stress generation and toque generation in rotating-field electrical machines under a heavy load.

(Pyrhönen, et al., 2014a)

Figure 12 illustrates the flux solution for an asynchronous machine showing the air-gap flux paths when the machine was heavily loaded. The flux lines can be seen such that they are inclined to produce tangential field strengths.

Figure 12 Flux solution of a loaded induction (asynchronous) machine showing large tangential field strength that produces a high torque. The enlarged figure on the left shows the tangential and normal field strength components.

(Pyrhönen, et al., 2014a) Reproduced by permission of Wiley

According to Maxwell’s stress theory, the magnetic field strength between the objects in vacuum creates a stress 𝜎_𝐹 on the object surfaces given by the following relation.

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36

𝜎_𝐹 = ¹₂𝜇₀𝐻² (3)

The stress also occurs in the direction of lines of force, which creates an equal pressure perpendicular to the lines. This stress term when resolved into its normal and tangential components yields the following two relations:

𝜎_𝐹n = ¹₂𝜇₀(𝐻_n²− 𝐻_tan² ) (4),

𝜎_𝐹tan = 𝜇₀𝐻_n𝐻_tan (5)

A linear current density creates A creates a tangential field strength in the machine such that 𝐻_tan,δ = 𝐴 and 𝐵_tan,δ = 𝜇₀ 𝐴. Thus, the tangential stress given by equation (5) becomes:

𝜎_Ftan = 𝜇₀𝐻_n𝐴 = 𝐵_n𝐴 (6)

The higher the tangential stress offered by the machine, the higher is the torque of the machine.

Equation (5) above shows that the tangential stress of the machine can be increased by increasing the magnetic flux density. In case of air-gap, the higher air-gap magnetic flux density can produce higher tangential stress and thus a higher torque.

The torque 𝑇 of the machine with rotor diameter of 𝐷_r, rotor surface area of 𝑆_r, and the tangential stress 𝜎_𝐹tan is thus given by equation (7).

𝑇 = 𝜎_𝐹tan^𝐷₂^r 𝑆_r (7)

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37

4. INVESTIGATIONS AND COMPUTATIONS

The machine investigated was a 12-pole permanent magnet synchronous motor. The laminations of the motor were constructed using steel material M270-50A whose typical BH -characteristic data and BH - characteristic curve, computed for frequency of 50 Hz can be found in APPENDIX I and APPENDIX II respectively. The investigations have been performed to find out the air-gap magnetic flux density and compute the torque production by the corresponding air-gap magnetic flux density. In addition, rotor iron loss and spectrum analysis of transient torque have also been performed for the U-shaped rotor geometry. Further computations of d- axis and q-axis inductances for both the U-shaped and dove-tail shaped rotors have been performed.

The investigation has been started computing the air-gap magnetic flux density for different heights of the magnet at no-load conditions for a magnet placed on rotor surface. In addition, a V-shaped arrangement of magnets has also been investigated at no-load condition. In these investigations, only the air-gap magnetic flux density has been considered without introducing currents into the coil conductors of the stator. Further, investigations have been carried out such that the coil conductors of the stator are supplied with currents. The magnitudes of the currents in each of the coil conductors of the three phases have been shown in Table 4.

Table 4 showing the supplied currents to the phases (+U, -U), (+V, -V), and (+W, -W). 𝑖̂ .is the amplitude of the supplied RMS current

Phase +U +V +W -U -V -W

Current [A] 1

2∙ 𝑖̂ − 𝑖̂ 1

2∙ 𝑖̂ −1

2∙ 𝑖̂ 𝑖̂ −1

2∙ 𝑖̂

Moreover, two different geometries of embedded magnets were investigated to find out the possibility to get a higher air-gap magnetic flux density so as to get a higher torque. One of the geometries had magnets arranged in dove-tail fashion and the other had magnets arranged in U- shape. The schematic diagrams of the two geometries are shown in Figure 13. In both cases the

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38 idea was to increase the magnet material surface per pole to create a larger flux density in the air gap.

Figure 13 Schematic diagram of magnets arranged in (a) dove-tail fashion and (b) U-shape

Further investigations were carried out in those two geometries altering the magnet arrangements slightly such that the geometry did not deviate from the original geometry. The aim of altering the geometry was to find out an optimized magnet arrangement in those two magnet arrangements. The magnetic field orientation in the U-shape arrangement was such that a Halbach array was produced.

4.1. Analysis with Different Magnet Heights

In this analysis, the variations of air-gap magnetic flux density with varying magnet heights in the rotor at no load were investigated. The magnet height was varied from 5 mm to 85 mm.

Figure 14 shows the schematic diagram of the magnet, and the arrows represent its direction of magnetic field orientation.

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39 Figure 14 showing the schematic diagram of a magnet block with the direction of polarization orientation represented by the arrows. The regions of stator, rotor, and the air-gap have also been labelled in the figure.

Figure 15 shows the schematic geometry of the rotor with embedded magnet employed in this investigation. The height of the magnet was increased along the dashed-red line.

Figure 15 showing the magnet arrangement embedded in the rotor. The figure shows the shaft, the rotor, the air- gap, and the stator regions along with the embedded magnet with parallel magnetization and its height. The physical air gap is 2 mm.

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40 Table 5 shows the computed air-gap magnetic flux densities with different heights of the magnet. It can be seen that the maximum computed air-gap magnetic flux density corresponds to a magnet height of 25 mm. However, the variations are not very significant with heights greater or less than 25 mm.

Table 5 showing the air-gap magnetic flux density against the depth of the permanent magnet with 2 mm air gap.

The remanence of the magnet material is Br = 1.22 T

Height of magnet [mm] Maximum air-gap magnetic flux density, B / [T]

5 0.80

10 0.94

15 0.99

25 1.00

35 0.99

45 0.97

55 0.96

65 0.96

75 0.95

85 0.95

Figure 16 illustrates the curve showing the variations of the air-gap magnetic flux densities with increasing magnet height. The curve shows the maximum achievable air-gap magnetic flux density with different heights of magnet. It can be seen in Figure 16 that the air-gap magnetic flux density saturates at magnet heights beyond 55 mm. A fitting curve has also been shown to represent the trend of air-gap magnetic flux density in the air-gap with increasing magnet height.

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41 Figure 16 showing the curve of maximum air-gap flux density with increasing height of magnet with 2 mm physical air gap. The blue solid curve represents the computed maximum air-gap magnetic flux density, while the dashed- red curve represents the best fitting curve (spline fitting) for the computed maximum air-gap magnetic flux density.

The Remanence of the magnet material is Br = 1.22 T

Furthermore, Figure 17 shows the magnetic flux density distributions for four different magnet heights of 5 mm, 10 mm, 15 mm, and 25 mm. The curves corresponding to magnet heights of 15 mm and 25 mm almost coincide. Therefore, it can also be concluded that magnet height in between 15 mm and 25 mm may be chosen to get a maximum air-gap magnetic flux density of approximately 1 tesla.

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42 Figure 17 Air-gap magnetic flux density distributions for magnet heights of 5 mm, 10 mm, 15 mm, and 25 mm with non-slotted air gap. The horizontal axis represents the distance of points on the curve defined in the air-gap, and the vertical axis shows the air-gap magnetic flux density. The Remanence of the magnet material is Br = 1.22 T and the physical air gap 2 mm.

4.2. Analysis with three-pieces of magnets

The above analysis was extended such that there were three magnet pieces connected end-to- end per pole. The magnetic polarization orientation of each piece of magnet is shown in Figure 18 with arrows. In this case, the widths of the two side magnet were shorter than that of the middle one.

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43 Figure 18 showing three magnet pieces and their polarization orientation. The figure also illustrates the stator, rotor, and air-gap regions.

The schematic rotor geometry with three magnet pieces is shown in Figure 19. The width of the middle magnet piece was set to be twice the widths of the two side magnet pieces. Finite Element Analysis (FEM) was then employed to investigate the air-gap magnetic flux density. In this case also, the magnet depths were varied along the dashed-red line as illustrated in Figure 19 below.

Figure 19 Three pieces of magnets embedded in the rotor. The figure also shows the shaft, rotor, air-gap (2 mm), and stator regions.

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44 The investigation result for the air-gap magnetic flux density is shown in Figure 20. The curves represent the air-gap magnetic flux density distributions for four different magnet heights of 10mm, 35 mm, 50 mm, and 70 mm. The curves clearly indicate that the maximum air-gap magnetic flux density could not be increased to a higher level even with a higher height of the magnet when they are arranged in the fashion as shown in Figure 19 above. Therefore, it would not be a wise choice to just increase the height of the magnet as it increases the machine weight and its construction cost.

Figure 20 Air-gap magnetic flux density curves for three different magnet heights, of 35 mm, 50 mm, and 70 mm.

Magnets are assembled according to Figure 19. The remanence of the magnet material is Br = 1.22 T and the physical air gap 2 mm.

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45 In addition, Table 6 shows the maximum, minimum, RMS, and the mean values of the air-gap magnetic flux densities for the curves shown in Figure 20 above.

Table 6 Maximum, minimum, mean, and RMS values of air-gap flux density corresponding to four different magnet heights of 10 mm, 35 mm, 50 mm, and 75 mm. The magnets are oriented according to Figure 18. The remanence of the magnet material is Br = 1.22 T and the physical air gap is 2 mm.

Height of magnet [mm]

Maximum air- gap magnetic flux density [T]

Minimum air- gap magnetic flux density [T]

Mean air-gap magnetic flux density [T]

RMS value of the air-gap magnetic flux density [T]

10 0.474 0.263 0.319 0.323

35 0.798 0.627 0.670 0.671

50 0.831 0.683 0.738 0.738

75 0.849 0.705 0.766 0.767

4.3. Analysis with magnets arranged in V-shape

Another rotor geometry in which the magnets were arranged in V-shape was investigated. For simplicity, Figure 21 shows the schematic diagram of the magnet pieces. Their magnetic field orientations have been represented by the arrows.

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46 Figure 21 showing the magnets arranged in V-shape and the direction of magnetic field orientation. The figure also illustrates the stator, rotor, and air-gap regions

As in earlier two cases, Figure 22 shows the schematic geometry of the rotor with magnets arranged in V-shape.

Figure 22 showing one pole of the rotor where two magnets are arranged in V-shape. The figure shows the shaft, rotor, air-gap (2 mm), and stator regions. The magnets are shown by red blocks.

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47 Figure 23 illustrates the curves for the air-gap magnetic flux density distributions with three different magnet heights with V-shape magnet arrangements. The curve for magnets with 5 mm magnet height depicts that the air-gap magnetic flux density distribution is greater in comparison to the magnets with 13.5 mm height. Moreover, the curve for magnets with the height of 9.5 mm shows the highest air-gap magnetic flux density distribution.

Because of geometrical constraints the magnet widths were taken different for different heights of the magnet. For example, the magnet width with magnet heights of 5 mm and 9.5 mm was about 27 mm while the magnet width of the magnets with height of 13.5 mm was about two- thirds of the width of the magnets with the height of 5 mm or 9.5 mm. This also signifies that the air-gap magnetic flux density cannot be just increased by increasing the height of the magnets when arranged in a V-shape. The width of the magnets also plays an important role. In addition to this, the angle at which the magnets are arranged in a V-shape also play an important role, because the angle of the magnets with 13.5 mm height was less than in the other two cases.

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48 Figure 23 showing the curves of air-gap magnetic flux densities and rotor geometries with magnets arranged in V- shape. The curves represent three different heights of magnets – 5 mm, 9.5 mm, and 13.5 mm. The remanence of the magnet is Br = 1.22 T and the air gap 2 mm.

After performing the no-load condition investigations for different magnet heights along with the V-shaped magnet, further investigations were performed on two rotor geometries with magnets arranged in dove-tail fashion or U-shape. As said earlier, the U-shape arrangement of magnets were oriented such that a Halbach-array-resembling magnet system was produced. A current corresponding to a per unit (pu) value of 1 was supplied to the coil conductors of the

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49 stator and further investigations were performed using finite element analysis (FEA) software.

However, the frequency converter supplying the investigated motor has an ability to supply a current corresponding to a per unit (pu) value of 1.4, thus this case has also been investigated.

For comparisons, further investigations with a current corresponding to a per unit (pu) value of 1.6 were also performed.

4.4. Analysis with U-shaped magnet arrangement

The magnetic polarizations of the magnets in the rotor with U-shaped magnet arrangements were oriented to produce a Halbach array resembling system. In this investigation, currents were supplied to the coil conductors of the stator to investigate the air-gap magnetic flux density distributions, and the torque the motor was able to generate. The torque production has been investigated with respect to the rotor angular positions with constant current supply. The air- gap magnetic flux density distribution has been investigated on different points on the curve defined in the air-gap of the machine.

Three different rotor geometries with Halbach arrangement resembling system of magnets have been investigated. The rotor geometries have been defined as Rotor Type A, Rotor Type B, and Rotor Type C. Rotor Type A was the original geometry while Rotor Type B and Rotor Type C were slight modifications to the original geometry such that the original geometry integrity did not alter and an optimized rotor geometry could be found. Table 7 shows the three rotor types with their corresponding geometries.The arrows on the magnets show their polarization direction.

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50 Table 7 Three different rotor types with magnets arranged in U-shape

Rotor Type Rotor Geometry

A

B

C

Figure 24 shows the machine geometry with Rotor Type A. The figure shows one complete pole and halves of two adjacent poles.